Optimal a priori error estimate of relaxation-type linear finite element method for nonlinear Schrödinger equation

Abstract

In this paper, we study, analyze and numerically validate a conservative relaxation-type linear finite element method (FEM) for the nonlinear Schrödinger equation. The method avoids solving the complex nonlinear system and preserves the discrete mass and energy. A key to our analysis is that the errors are split into the temporal error and the spatial error by introducing the corresponding time-discrete system. Therefore, we derive the optimal L2 and semi-H1 error estimates without any coupling condition between time step τ and space size h. Some numerical experiments not only demonstrate the method’s optimal convergence rates but also confirm the conservation laws during long time simulations.